The tags/locations on the map would be clickable, either on the map itself or on the list to the lift. (Appearance and detail will, of course, vary in the final solution.) Exploring a hex takes you to a turn-based exploration map. Each hex would have a seed based on the world seed combined with the hex coordinates, so that on each visit the same location would be generated. I'm still pondering whether to make the sub-maps be persistent or not; my natural inclination is to re-roll the contents each exploration and only make events persistent. Events can be attached to a hex, such as the Landing Area, Hills of Alfar and Pirate Camp events above. These generate an encounter on top of the generated hex location. So while any hex in the map can be explored, exploring a hex with an event attached takes you to a 'special' instance of the location, with all the event goodies in place. For instance, exploring the Landing Area event would generate the shoreline map with a usable object representing your raft which would allow you to sail away to another island. This location would be a good place to construct your base.
Of course, this is only a mockup, so the final appearance will probably vary. I'm sort of toying with the idea of the world map being accessible from a special type of workbench in your base location. Your first order of business upon landing on a new island would be to perform a sort of magical survey to generate the map representation. I'd probably just abstract this away to the action "Build Map Table", after which you'd have a workbench you could access to get to the map. (Hence the above representation of the map as a laid-out piece of parchment on a wood surface.)
On first arrival, only a small handful of events near the landing area are available. As you complete events, others unlock, expanding across the map and culminating in the final 'boss' encounter of the map. (Which might not be a boss, per se, but rather just some event that represents the point or theme of the island.) Some events represent the main exploration thread of an island, others are just simple random encounters that don't necessarily unlock further events.
In the mockup above, there is no world map travel. It's simple tag selection to open a dialog, "Explore this location?" sort of thing. It is then assumed that all hexes on the map are reachable, without pathfinding troubles, and progress is locked behind the progression of event exploration.